Abstract
We propose some simplified modifications of the partial linearization method for network equilibrium problems with mixed demand. In these modifications, the auxiliary direction choice problem is solved approximately. In the modifications, the basic convergence properties of the original method are preserved, while the inexact solution of the auxiliary problems reduces the computational efforts. Preliminary numerical tests show the advantages and efficiency of our approach as compared with the exact variant of the method.
This is a preview of subscription content, log in via an institution to check access.
References
S. Dafermos, “Traffic Equilibrium and Variational Inequalities,” Transp. Sci. 14 (1), 42–54 (1980).
S. Dafermos, “The General Multimodal Network Equilibrium Problem with Elastic Demand,” Networks 12 (1), 57–72 (1982).
A. Nagurney, Network Economics: A Variational Inequality Approach (Kluwer Acad. Publ., Dordrecht, 1999).
M. Patriksson, The Traffic Assignment Problem: Models and Methods (Dover, Mineola, NY, 2015).
T. L. Magnanti, “Models and Algorithms for Predicting Urban Traffic Equilibria,” in Transportation Planning Models, ed. by M. Florian (North-Holland, Amsterdam, 1984), pp. 153–185.
I. Konnov and O. Pinyagina, “Partial Linearization Method for Network Equilibrium Problems with Elastic Demands,” in Discrete Optimization and Operations Research (Proceedings of 9th International Conference DOOR-2016, Vladivostok, Russia, September 19–23, 2016) (Springer, Heidelberg, 2016), pp. 418–429.
I. V. Konnov, “Simplified Versions of the Conditional Gradient Method,” Optimization 67 (12), 2275–2290 (2018).
H. Mine and M. Fukushima, “A Minimization Method for the Sum of a Convex Function and a Continuously Differentiable Function,” J. Optim. Theor. Appl. 33, 9–23 (1981).
M. Patriksson, “Cost Approximation: A Unified Framework of Descent Algorithms for Nonlinear Programs,” SIAM J. Optim. 8, 561–582 (1998).
M. Patriksson, Nonlinear Programming and Variational Inequality Problems: A Unified Approach (Kluwer Acad. Publ., Dordrecht, 1999).
K. Bredies, D. A. Lorenz, and P. Maass, “A Generalized Conditional Gradient Method and Its Connection to an Iterative Shrinkage Method,” Comput. Optim. Appl. 42, 173–193 (2009).
G. Scutari, F. Facchinei, P. Song, D. P. Palomar, and J.-S. Pang, “Decomposition by Partial Linearization: Parallel Optimization of Multi-Agent Systems,” IEEE Trans. Signal Process 62, 641–656 (2014).
I. V. Konnov, “On Auction Equilibrium Models with Network Applications,” NETNOMICS: Economic Research and Electronic Networking 16 (1), 107–125 (2015).
O. Pinyagina, “On a Network Equilibrium Problem with Mixed Demand,” Discrete Optimization and Operations Research (Proceedings of 9th International Conference DOOR-2016, Vladivostok, Russia, September 19–23, 2016) (Springer, Heidelberg, 2016), pp. 578–583.
O. V. Pinyagina, “The Network Equilibrium Problem with Mixed Demand,” Diskretn. Anal. Issled. Oper. 24 (4), 77–94 (2017)
O. V. Pinyagina, J. Appl. Indust. Math. 11 (4), 554–563 (2017)].
D. P. Bertsekas and E. M. Gafni, “Projection Methods for Variational Inequalities with Application to the Traffic Assignment Problem,” in Nondifferential and Variational Techniques in Optimization (Springer, Heidelberg, 1982), pp. 139–159.
A. B. Nagurney, “Comparative Tests of Multimodal Traffic Equilibrium Methods,” Transportation Research Part B: Methodological 18 (1), 469–485 (1984).
Funding
The first author was supported by the Russian Foundation for Basic Research (project no. 19-01-00431) and the Ministry of Science and Education of Russia (State Assignment no. 1.460.2016/1.4). The first and second authors were supported by the Academy of Finland (Grant no. 315471).
Author information
Authors and Affiliations
Corresponding authors
Additional information
Russian Text © The Author(s), 2020, published in Diskretnyi Analiz i Issledovanie Operatsii, 2020, Vol. 27, No. 1, pp. 43–60.
Rights and permissions
About this article
Cite this article
Konnov, I.V., Laitinen, E. & Pinyagina, O.V. Inexact Partial Linearization Methods for Network Equilibrium Problems. J. Appl. Ind. Math. 14, 92–103 (2020). https://doi.org/10.1134/S199047892001010X
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1134/S199047892001010X